366 



macfarlane: algebra of physics 



F-/p 4 



Under the first heading are considered two 

 lines OA and OP (fig. 4) with the projections 

 of OP along and perpendicular to OA, namely 

 OM and MP, also certain other projections 

 as AT drawn perpendicular to OA cutting off 

 OT. It was shown that the geometric equa- 

 tion of the first degree 



OP = OM + MP 



leads to a geometric equation of the second 

 degree 



(OA) (OP) = (OA) (OM) + (OA) (MP) ; 



that is, the parallelogram (OA) (OP) is equal to the scalar area 

 (OA) (OM) plus the vector area (OA) (MP). This was stated to 

 be the fundamental principle of vector-analysis. The complete 

 product (whose existence is ignored by vector-analysts) is simply 

 the parallelogram formed by the two given lines OA and OP. Its 

 unit, on account of its obliquity partakes partly of the nature of 

 the scalar unit, and partly of the nature of the orthogonal unit. 

 The fundamental principle is expressed by 



OA) (OP) = (OA) (OM) + (OA) (MP) (fig. 5.) 



o fl 



In a similar manner, as OA = OT — AT, it follows that 

 (OA) (OA) = (OA) (OT) - (OA) (AT) 

 It was pointed out that there is a variety of ways of defining the 



